Temporary HW

Problem 1

The Hamiltonian for an electron spin degree of freedom in the external magnetic field $\mathbf {B}$ is given by:

${\hat {H}}=-\mu _{B}{\vec {\sigma }}\cdot \mathbf {B}$

where $\mu _{B}istheBohrmagneton$ and ${\vec {\sigma }}=({\hat {\sigma }}_{x},{\hat {\sigma }}_{y},{\hat {\sigma }}_{z})$ is the vector of the Pauli matrices:

${\hat {\sigma }}_{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},$

${\hat {\sigma }}_{x}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},$

${\hat {\sigma }}_{x}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.$

(a) In the quantum canonical ensemble, evaluate the density matrix if $\mathbf {B}$ is along the z axis.

(b) Repeat the calculation from (a) assuming that $\mathbf {B}$ points along the x axis.

(c) Calculate the average energy in each of the above cases.